05%20-%20Ellipse

05%20-%20Ellipse - PROBLEMS 05 ELLIPSE Page 1 1 The...

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PROBLEMS 05 - ELLIPSE Page 1 ( 1 ) The end-points A and B of AB are on the X- and Y-axis respectively. If AB = a + b, a > 0, b > 0, a b and P divides AB from A in the ratio b : a, then show that P lies on the ellipse 2 2 2 2 b y a x + = 1. ( 2 ) If the feet of the perpendiculars drawn to the tangent at any point of the ellipse 2 2 2 2 b y a x + = 1 from foci S and S’ are L and L’ respectively, then show that SL L' S' = b 2 . ( 3 ) Prove that the line segment of any tangent, between the tangents at the end-points of the major axis, forms a right angle at either focus of the ellipse. ( 4 ) Show that the equation of the chord joining the points P ( α ) and Q ( β ) of the ellipse 2 2 2 2 b y a x + = 1 is 2 β α cos 2 β α sin b y 2 β α cos a x - = + + + . ( 5 ) If the chord joining the points P ( α ) and Q ( β ) of the ellipse 2 2 2 2 b y a x + = 1 passes through the focus ( ae, 0 ), then prove that 1 e 1 e 2 β tan 2 α tan + = - . ( 6 ) If the chord joining the points P ( α ) and Q ( β ) of the ellipse 2 2 2 2 b y a x + = 1 subtends a right angle at the centre, then show that tan α β tan + 2 2 b a = 0 and if it forms a right angle at the vertex ( a, 0 ), then show that 0 a b 2 β tan 2 α tan 2 2 = + . ( 7 ) If the difference of eccentric angles of the points P and Q on the ellipse 2 2 2 2 b y a x + = 1 is 2 π and PQ cuts intercepts of length c and d on the axes, then prove that 2 d b c a 2 2 2 2 = + .
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PROBLEMS 05 - ELLIPSE Page 2 ( 8 ) If two radii CP and CQ of the ellipse 2 2 2 2 b y a x + = 1 are perpendicular, then prove that 2 2 CQ 1 CP 1 + = 2 2 b 1 a 1 + , where C is the centre
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This note was uploaded on 11/28/2011 for the course MATH 300 taught by Professor Jones during the Spring '06 term at ITT Tech Flint.

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05%20-%20Ellipse - PROBLEMS 05 ELLIPSE Page 1 1 The...

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